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Asymptotic behavior of second-order nonlinear neutral dynamic equations
Journal of Inequalities and Applications volume 2013, Article number: 505 (2013)
Abstract
This paper is concerned with oscillation and asymptotic behavior of a second-order neutral delay dynamic equation on an arbitrary time scale. We obtain two theorems which guarantee that every solution of the studied equation oscillates or converges to zero. These results improve and complement some known results given in the literature.
MSC:34K11, 34N05, 39A10, 39A12, 39A13, 39A21.
1 Introduction
In this paper, we study oscillation and asymptotic behavior of a second-order nonlinear neutral delay dynamic equation
on an arbitrary time scale , where γ is a quotient of odd positive integers, r and p are positive rd-continuous functions on , . Also, we assume that are rd-continuous, , , , for all , and there exists a positive rd-continuous function q defined on such that .
The theory of dynamic equations on time scales, which goes back to its founder Hilger [1], has recently attracted attention of researchers. Several authors have expounded on various aspects of this new theory; see the survey paper written by Agarwal et al. [2] and the references cited therein. The books on the subject of time scales, by Bohner and Peterson [3, 4], present much of time scale calculus.
Since we are interested in oscillatory and asymptotic properties, we assume throughout this paper that the given time scale is unbounded above. We assume that , and it is convenient to assume that , and define the time scale interval of the form by . Throughout, we use the notation . By a solution of equation (1.1), we mean a non-trivial real-valued function , which has the property that z and are defined and Δ-differentiable for and satisfies equation (1.1) on . The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory.
In recent years, there has been much research activity concerning oscillation and nonoscillation of solutions to neutral differential and dynamic equations on time scales, we refer the reader to [5, 6] and [7–22], and the references cited therein. Han et al. [6] studied a second-order nonlinear neutral equation
where , and established two results which guarantee that every solution of equation (1.2) is oscillatory under the assumptions that
and
Agarwal et al. [7], Erbe et al. [8], Şahiner [9], Saker [10], Saker et al. [11], Saker and O’Regan [12], Chen [13], Zhang and Wang [14], Wu et al. [15], Candan [17], and Li et al. [19] investigated equation (1.1) and obtained some oscillation criteria in the case
As yet, there are few results regarding the study of asymptotic behavior of equation (1.1) under the assumption that
In 2007, Saker et al. [11] posed an open problem as follows: How to establish oscillation criteria for equation (1.1) when condition (1.5) holds? Assuming (1.5), Zhang et al. [21, 22] obtained some sufficient conditions which insure that all solutions of equation (1.1) are oscillatory.
The purpose of this paper is to present some asymptotic tests for equation (1.1) in the case where (1.5) holds. This paper is organized as follows: In the next section, we shall establish the main results. In Section 3, two examples are provided to illustrate the results obtained.
In the sequel, when we write a functional inequality without specifying its domain of validity, we assume that it holds for all sufficiently large t.
2 Main results
In what follows, we use the notation
and, for sufficiently large , .
In order to prove our main results, we will use the following result; see [[8], Theorem 2.1].
Theorem 2.1 Let (1.4) hold. Suppose that there exists a positive Δ-differentiable function δ such that for all sufficiently large and for ,
Then every solution x of equation (1.1) is oscillatory.
Theorem 2.2 Let (1.5) hold. Assume that there exists a positive Δ-differentiable function δ such that for all sufficiently large and for , one has (2.1). If
then every solution x of equation (1.1) is oscillatory or .
Proof Let x be a nonoscillatory solution of equation (1.1). Without loss of generality, we assume that , , and for . Then for . In view of (1.1), we get
Therefore, is strictly decreasing, and there exists a such that or for . We consider each of two cases separately.
Case 1. Assume that for . As in the proof of [[8], Theorem 2.1], we can obtain a contradiction to (2.1).
Case 2. Assume that for . Then, there exists a finite limit
where . Now, we claim that . If not, then for any , we have , eventually. Take . We calculate
where
Since is strictly decreasing,
Integrating the inequality above from t to l and letting , we have by (2.3) that
where . Combining (2.4) and (2.5), we get
Then by (2.3), we obtain
Integrating the inequality above from () to t, we have
which implies that
Integrating the latter inequality from to t, we get
which yields , this is a contradiction. Hence, . By virtue of , . The proof is complete. □
Next, we establish another criterion which improves Theorem 2.2.
Theorem 2.3 Let (1.5) hold. Suppose that there exists a positive Δ-differentiable function δ such that for all sufficiently large and for , one has (2.1). If
then every solution x of equation (1.1) is oscillatory or .
Proof Let x be a nonoscillatory solution of equation (1.1). Without loss of generality, we assume that , , and for . Then for . In view of (1.1), we get (2.3). Thus, is strictly decreasing, and there exists a such that or for . We consider each of two cases separately.
Case 1. Assume that for . Similarly to the proof of [[8], Theorem 2.1], we can obtain a contradiction to (2.1).
Case 2. Assume that for . Then there exists a finite limit
where . Next, we claim that . If not, then for any , we have , eventually. Take . Then we have (2.4). It follows from (2.3), (2.4), and that
Integrating the inequality above from () to t, we get
which yields
Integrating the latter inequality from to t, we have
which implies that , this is a contradiction. Hence, . By , . This completes the proof. □
Remark 2.1 When , Theorems 2.2 and 2.3 improve results of Han et al. [[6], Theorems 2.1 and 2.2] since our results do not require condition (1.3).
Remark 2.2 The results obtained in this paper complement the recent results given in [7–19] in the sense that these results can be applied to case (1.5).
3 Applications
In this section, we give two examples to illustrate applications of results in the previous section.
Example 3.1 For , consider a second-order neutral delay dynamic equation
where satisfying , , , and . Let and . Then, we have
that is, (1.5) holds. Note that , and for every constant and for ,
Choose . It is not difficult to verify that (2.1) holds. On the other hand,
Thus, we have by Theorem 2.2 that every solution x of (3.1) is oscillatory or .
Example 3.2 For , consider a second-order neutral delay dynamic equation
where satisfying , , , and . Let . Then we have
that is, (1.5) holds. Note that , and for every constant and for ,
Choose . It is easy to verify that (2.1) holds. On the other hand,
Hence, by Theorem 2.3, every solution x of (3.2) is oscillatory or .
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Acknowledgements
This research is supported by the National Key Basic Research Program of P.R. China (2013CB035604) and the NNSF of P.R. China (Grant Nos. 61034007, 51277116, and 51107069).
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Jing, Y., Zhang, C. & Li, T. Asymptotic behavior of second-order nonlinear neutral dynamic equations. J Inequal Appl 2013, 505 (2013). https://doi.org/10.1186/1029-242X-2013-505
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DOI: https://doi.org/10.1186/1029-242X-2013-505